\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  23.0 & -7.00 x_{1} & -4.00 x_{2} & -1.00 x_{3} & -8.00 x_{4} & -10.00 x_{5} & -7.00 x_{6}\\
 x_{8}   &  5.0 & +  3.00 x_{1} & +  8.00 x_{2} & +  5.00 x_{3} & +  8.00 x_{4} & -3.00 x_{5} & -8.00 x_{6}\\
 x_{9}   &  7.0 & +  4.00 x_{1} & +  2.00 x_{2} & +  8.00 x_{3} & -5.00 x_{4} & -6.00 x_{5} & +  1.00 x_{6}\\
 x_{10}   &  7.0 & +  9.00 x_{1} & -2.00 x_{2} & -5.00 x_{3} & +  3.00 x_{4} & +  1.00 x_{5} & -7.00 x_{6}\\
 x_{11}   &  -11.0 & +  9.00 x_{1} &   & -7.00 x_{3} & -7.00 x_{4} & +  9.00 x_{5} & -9.00 x_{6}\\
 x_{12}   &  -16.0 & -1.00 x_{1} & +  8.00 x_{2} & +  9.00 x_{3} & -4.00 x_{4} & +  4.00 x_{5} & +  7.00 x_{6}\\
 x_{13}   &  -1.0 & -2.00 x_{1} & +  5.00 x_{2} & -6.00 x_{3} & +  1.00 x_{4} & +  6.00 x_{5} & -9.00 x_{6}\\
\hline
z    &  0.0 & +  5.00 x_{1} & -3.00 x_{2} & -5.00 x_{3} & +  5.00 x_{4} & +  5.00 x_{5} & -3.00 x_{6}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{6}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  7.00 y_{7} & -3.00 y_{8} & -4.00 y_{9} & -9.00 y_{10} & -9.00 y_{11} & +  1.00 y_{12} & +  2.00 y_{13}\\
 y_{2}   &  1.0 & +  4.00 y_{7} & -8.00 y_{8} & -2.00 y_{9} & +  2.00 y_{10} &   & -8.00 y_{12} & -5.00 y_{13}\\
 y_{3}   &  1.0 & +  1.00 y_{7} & -5.00 y_{8} & -8.00 y_{9} & +  5.00 y_{10} & +  7.00 y_{11} & -9.00 y_{12} & +  6.00 y_{13}\\
 y_{4}   &  1.0 & +  8.00 y_{7} & -8.00 y_{8} & +  5.00 y_{9} & -3.00 y_{10} & +  7.00 y_{11} & +  4.00 y_{12} & -1.00 y_{13}\\
 y_{5}   &  1.0 & + 10.00 y_{7} & +  3.00 y_{8} & +  6.00 y_{9} & -1.00 y_{10} & -9.00 y_{11} & -4.00 y_{12} & -6.00 y_{13}\\
 y_{6}   &  1.0 & +  7.00 y_{7} & +  8.00 y_{8} & -1.00 y_{9} & +  7.00 y_{10} & +  9.00 y_{11} & -7.00 y_{12} & +  9.00 y_{13}\\
\hline
z    &  -0 & -23.00 y_{7} & -5.00 y_{8} & -7.00 y_{9} & -7.00 y_{10} & + 11.00 y_{11} & + 16.00 y_{12} & +  1.00 y_{13}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 4 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{11}   &  0.0555555555556 & +  0.89 y_{7} & +  0.78 y_{8} & +  0.78 y_{9} & -0.22 y_{10} & +  0.06 y_{2} & -0.11 y_{5} & -0.39 y_{13}\\
 y_{1}   &  0.625 & -0.50 y_{7} & -11.00 y_{8} & -11.25 y_{9} & -6.75 y_{10} & -0.62 y_{2} & +  1.00 y_{5} & +  4.88 y_{13}\\
 y_{3}   &  0.263888888889 & +  2.72 y_{7} & +  9.44 y_{8} & -0.31 y_{9} & +  1.19 y_{10} & +  1.51 y_{2} & -0.78 y_{5} & +  8.90 y_{13}\\
 y_{4}   &  1.88888888889 & + 16.22 y_{7} & -6.56 y_{8} & +  9.44 y_{9} & -3.56 y_{10} & -0.11 y_{2} & -0.78 y_{5} & -6.22 y_{13}\\
 y_{12}   &  0.125 & +  0.50 y_{7} & -1.00 y_{8} & -0.25 y_{9} & +  0.25 y_{10} & -0.12 y_{2} &   & -0.62 y_{13}\\
 y_{6}   &  0.625 & + 11.50 y_{7} & + 22.00 y_{8} & +  7.75 y_{9} & +  3.25 y_{10} & +  1.38 y_{2} & -1.00 y_{5} & +  9.88 y_{13}\\
\hline
z    &  2.61111111111 & -5.22 y_{7} & -12.44 y_{8} & -2.44 y_{9} & -5.44 y_{10} & -1.39 y_{2} & -1.22 y_{5} & -13.28 y_{13}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  5.22222222222 & -0.89 x_{11} & +  0.50 x_{1} & -2.72 x_{3} & -16.22 x_{4} & -0.50 x_{12} & -11.50 x_{6}\\
 x_{8}   &  12.4444444444 & -0.78 x_{11} & + 11.00 x_{1} & -9.44 x_{3} & +  6.56 x_{4} & +  1.00 x_{12} & -22.00 x_{6}\\
 x_{9}   &  2.44444444444 & -0.78 x_{11} & + 11.25 x_{1} & +  0.31 x_{3} & -9.44 x_{4} & +  0.25 x_{12} & -7.75 x_{6}\\
 x_{10}   &  5.44444444444 & +  0.22 x_{11} & +  6.75 x_{1} & -1.19 x_{3} & +  3.56 x_{4} & -0.25 x_{12} & -3.25 x_{6}\\
 x_{2}   &  1.38888888889 & -0.06 x_{11} & +  0.62 x_{1} & -1.51 x_{3} & +  0.11 x_{4} & +  0.12 x_{12} & -1.38 x_{6}\\
 x_{5}   &  1.22222222222 & +  0.11 x_{11} & -1.00 x_{1} & +  0.78 x_{3} & +  0.78 x_{4} &   & +  1.00 x_{6}\\
 x_{13}   &  13.2777777778 & +  0.39 x_{11} & -4.88 x_{1} & -8.90 x_{3} & +  6.22 x_{4} & +  0.62 x_{12} & -9.88 x_{6}\\
\hline
z    &  -2.61111111111 & -0.06 x_{11} & -0.62 x_{1} & -0.26 x_{3} & -1.89 x_{4} & -0.12 x_{12} & -0.62 x_{6}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  5.22222222222 & -0.89 x_{11} & +  0.50 x_{1} & -2.72 x_{3} & -16.22 x_{4} & -0.50 x_{12} & -11.50 x_{6}\\
 x_{8}   &  12.4444444444 & -0.78 x_{11} & + 11.00 x_{1} & -9.44 x_{3} & +  6.56 x_{4} & +  1.00 x_{12} & -22.00 x_{6}\\
 x_{9}   &  2.44444444444 & -0.78 x_{11} & + 11.25 x_{1} & +  0.31 x_{3} & -9.44 x_{4} & +  0.25 x_{12} & -7.75 x_{6}\\
 x_{10}   &  5.44444444444 & +  0.22 x_{11} & +  6.75 x_{1} & -1.19 x_{3} & +  3.56 x_{4} & -0.25 x_{12} & -3.25 x_{6}\\
 x_{2}   &  1.38888888889 & -0.06 x_{11} & +  0.62 x_{1} & -1.51 x_{3} & +  0.11 x_{4} & +  0.12 x_{12} & -1.38 x_{6}\\
 x_{5}   &  1.22222222222 & +  0.11 x_{11} & -1.00 x_{1} & +  0.78 x_{3} & +  0.78 x_{4} &   & +  1.00 x_{6}\\
 x_{13}   &  13.2777777778 & +  0.39 x_{11} & -4.88 x_{1} & -8.90 x_{3} & +  6.22 x_{4} & +  0.62 x_{12} & -9.88 x_{6}\\
\hline
z    &  1.94444444444 & +  0.72 x_{11} & -1.88 x_{1} & +  3.43 x_{3} & +  8.56 x_{4} & -0.38 x_{12} & +  6.12 x_{6}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  5.22222222222 & -0.89 x_{11} & +  0.50 x_{1} & -2.72 x_{3} & -16.22 x_{4} & -0.50 x_{12} & -11.50 x_{6}\\
 x_{8}   &  12.4444444444 & -0.78 x_{11} & + 11.00 x_{1} & -9.44 x_{3} & +  6.56 x_{4} & +  1.00 x_{12} & -22.00 x_{6}\\
 x_{9}   &  2.44444444444 & -0.78 x_{11} & + 11.25 x_{1} & +  0.31 x_{3} & -9.44 x_{4} & +  0.25 x_{12} & -7.75 x_{6}\\
 x_{10}   &  5.44444444444 & +  0.22 x_{11} & +  6.75 x_{1} & -1.19 x_{3} & +  3.56 x_{4} & -0.25 x_{12} & -3.25 x_{6}\\
 x_{2}   &  1.38888888889 & -0.06 x_{11} & +  0.62 x_{1} & -1.51 x_{3} & +  0.11 x_{4} & +  0.12 x_{12} & -1.38 x_{6}\\
 x_{5}   &  1.22222222222 & +  0.11 x_{11} & -1.00 x_{1} & +  0.78 x_{3} & +  0.78 x_{4} &   & +  1.00 x_{6}\\
 x_{13}   &  13.2777777778 & +  0.39 x_{11} & -4.88 x_{1} & -8.90 x_{3} & +  6.22 x_{4} & +  0.62 x_{12} & -9.88 x_{6}\\
\hline
z    &  1.94444444444 & +  0.72 x_{11} & -1.88 x_{1} & +  3.43 x_{3} & +  8.56 x_{4} & -0.38 x_{12} & +  6.12 x_{6}\\
\end{array}\]


 $ x_{3} $ enters and $ x_{2} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  2.7247706422 & -0.79 x_{11} & -0.62 x_{1} & +  1.80 x_{2} & -16.42 x_{4} & -0.72 x_{12} & -9.03 x_{6}\\
 x_{8}   &  3.77981651376 & -0.43 x_{11} & +  7.10 x_{1} & +  6.24 x_{2} & +  5.86 x_{4} & +  0.22 x_{12} & -13.42 x_{6}\\
 x_{9}   &  2.7247706422 & -0.79 x_{11} & + 11.38 x_{1} & -0.20 x_{2} & -9.42 x_{4} & +  0.28 x_{12} & -8.03 x_{6}\\
 x_{10}   &  4.34862385321 & +  0.27 x_{11} & +  6.26 x_{1} & +  0.79 x_{2} & +  3.47 x_{4} & -0.35 x_{12} & -2.17 x_{6}\\
 x_{3}   &  0.917431192661 & -0.04 x_{11} & +  0.41 x_{1} & -0.66 x_{2} & +  0.07 x_{4} & +  0.08 x_{12} & -0.91 x_{6}\\
 x_{5}   &  1.93577981651 & +  0.08 x_{11} & -0.68 x_{1} & -0.51 x_{2} & +  0.83 x_{4} & +  0.06 x_{12} & +  0.29 x_{6}\\
 x_{13}   &  5.11009174312 & +  0.72 x_{11} & -8.55 x_{1} & +  5.88 x_{2} & +  5.57 x_{4} & -0.11 x_{12} & -1.79 x_{6}\\
\hline
z    &  5.09174311927 & +  0.60 x_{11} & -0.46 x_{1} & -2.27 x_{2} & +  8.81 x_{4} & -0.09 x_{12} & +  3.01 x_{6}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{7} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{4}   &  0.165921787709 & -0.05 x_{11} & -0.04 x_{1} & +  0.11 x_{2} & -0.06 x_{7} & -0.04 x_{12} & -0.55 x_{6}\\
 x_{8}   &  4.75251396648 & -0.71 x_{11} & +  6.88 x_{1} & +  6.88 x_{2} & -0.36 x_{7} & -0.04 x_{12} & -16.64 x_{6}\\
 x_{9}   &  1.16145251397 & -0.34 x_{11} & + 11.73 x_{1} & -1.23 x_{2} & +  0.57 x_{7} & +  0.69 x_{12} & -2.85 x_{6}\\
 x_{10}   &  4.92402234637 & +  0.10 x_{11} & +  6.13 x_{1} & +  1.17 x_{2} & -0.21 x_{7} & -0.50 x_{12} & -4.07 x_{6}\\
 x_{3}   &  0.929608938547 & -0.04 x_{11} & +  0.41 x_{1} & -0.65 x_{2} & -0.00 x_{7} & +  0.08 x_{12} & -0.95 x_{6}\\
 x_{5}   &  2.07430167598 & +  0.04 x_{11} & -0.71 x_{1} & -0.42 x_{2} & -0.05 x_{7} & +  0.03 x_{12} & -0.17 x_{6}\\
 x_{13}   &  6.03407821229 & +  0.45 x_{11} & -8.76 x_{1} & +  6.49 x_{2} & -0.34 x_{7} & -0.36 x_{12} & -4.85 x_{6}\\
\hline
z    &  6.5530726257 & +  0.17 x_{11} & -0.79 x_{1} & -1.30 x_{2} & -0.54 x_{7} & -0.48 x_{12} & -1.83 x_{6}\\
\end{array}\]


 $ x_{11} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{11}   &  3.45348837209 & -20.81 x_{4} & -0.79 x_{1} & +  2.28 x_{2} & -1.27 x_{7} & -0.92 x_{12} & -11.44 x_{6}\\
 x_{8}   &  2.29069767442 & + 14.84 x_{4} & +  7.44 x_{1} & +  5.26 x_{2} & +  0.55 x_{7} & +  0.62 x_{12} & -8.49 x_{6}\\
 x_{9}   &  3.10862446895e-15 & +  7.00 x_{4} & + 12.00 x_{1} & -2.00 x_{2} & +  1.00 x_{7} & +  1.00 x_{12} & +  1.00 x_{6}\\
 x_{10}   &  5.26744186047 & -2.07 x_{4} & +  6.05 x_{1} & +  1.40 x_{2} & -0.34 x_{7} & -0.59 x_{12} & -5.21 x_{6}\\
 x_{3}   &  0.790697674419 & +  0.84 x_{4} & +  0.44 x_{1} & -0.74 x_{2} & +  0.05 x_{7} & +  0.12 x_{12} & -0.49 x_{6}\\
 x_{5}   &  2.22093023256 & -0.88 x_{4} & -0.74 x_{1} & -0.33 x_{2} & -0.10 x_{7} & -0.01 x_{12} & -0.65 x_{6}\\
 x_{13}   &  7.58139534884 & -9.33 x_{4} & -9.12 x_{1} & +  7.51 x_{2} & -0.91 x_{7} & -0.77 x_{12} & -9.98 x_{6}\\
\hline
z    &  7.1511627907 & -3.60 x_{4} & -0.93 x_{1} & -0.91 x_{2} & -0.76 x_{7} & -0.64 x_{12} & -3.81 x_{6}\\
\end{array}\]
Final Dictionary
Solution:  7.1511627907
Num Pivots:  4
\end{document}
